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doi:10.1016/j.ph

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Physica B 403 (2008) 3009–3012

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Are deformed modes still Raman activefor single-wall carbon nanotubes?

Wei Yanga, Ru-Zhi Wanga,�, Yu-Fang Wangb, Hui Yana

aLaboratory of Thin Film Materials, Beijing University of Technology, Beijing 100022, ChinabDepartment of Physics, Nankai University, Tianjin 300071, China

Received 16 October 2007; received in revised form 20 February 2008; accepted 3 March 2008

Abstract

Using group theory calculations combined with the force-constant model and molecular-dynamics simulations, whether the deformed

modes are Raman active or not for (10,10) and (10,0) single-wall carbon nanotubes under hydrostatic pressure is discussed. With

increasing pressure, the tube symmetry lowers from D20h to D2h then to C2h point group. For D20h, A1g is changed to Ag, while E1g and

E2g are split: E1g-B2g+B3g-2Bg and E2g-Ag+B1g-2Ag. On the basis of the correlation between D20h, D2h, and C2h point groups,

the deformed modes should be still Raman active. The result can help us clarify the essence of the experimental observations.

r 2008 Elsevier B.V. All rights reserved.

PACS: 61.46.+w; 62.50.+p; 63.22.+m

Keywords: Raman active; Symmetry; Correlation; Single-wall carbon nanotubes (SWCNTs); Hydrostatic pressure

Vibrational properties of carbon nanotubes (CNTs) areof considerable importance not only in the basic interestfor the phonons [1] but also in a number of nanomecha-nical devices [2]. Recently, the vibrational spectra at highpressure probed by Raman spectroscopy have attracted agreat deal of attention [3–9], mainly focusing on certainRaman-active vibrational modes, e.g., radial breathingmode (RBM) and tangential modes (TMs). Since the RBMand TMs are sensitive to the structural deformation ofCNTs under hydrostatic pressure, a disappearance of the Rband and a change in the pressure derivative of the T bandwere generally interpreted as a sign of structural transition[4–8]. However, theoretically, the RBM undergoes atransition around a critical pressure where a structuretransition occurs, and still exists at much higher pressurefor single-wall carbon nanotubes (SWCNTs) [10]. Ob-viously, once the tube structure is transformed, the modes,especially in terms of RBM, will be certainly affected, andeven the selection rules may be changed. For instance,

e front matter r 2008 Elsevier B.V. All rights reserved.

ysb.2008.03.006

ng author. Tel./fax: +86 01067392412.

ss: wrz@bjut.edu.cn (R.-Z. Wang).

symmetry alterations would remove some of the degen-eracies of the vibrational wave functions [11]. If the modesare located at Raman forbidden band, they will be nolonger Raman active, and thus experimentally undetectedby Raman spectra. Therefore, at much higher pressures,whether the deformed modes are Raman active or notwould be of particular important for catching on theessence of RBM change with radial compression. In thispaper, combined with the force-constant model [12] andmolecular-dynamics (MD) simulations [13], we presentgroup theoretical arguments on a series of structuralsymmetries with increasing hydrostatic pressure.In our calculations, (10,10) armchair and (10,0) zigzag

SWNTs are modeled by a supercell with 24 A width alongthe radial direction under a free boundary condition andfive unit cells along the axial direction under a periodboundary condition. In addition, the interactions betweencarbon atoms are obtained by an empirical Tersoff–Bren-ner many-body potential [14,15]. For the force-constantmodel, as defined in our previous work [10], a 3N� 3N (Nis the number of carbon atoms in the unit cell) dynamicsmatrix is calculated totally other than performing rotations

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600

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1400

1600

0.0 0.8 1.6

ω (c

m-1

)

(10, 10)

qq VDOS(arb.units)

VDOS(arb.units)

(10, 0)

1 2 3

Fig. 1. Calculated phonon dispersion relations and phonon density of

states for (a) (10,10) armchair and (b) (10,0) zigzag SWNTs.

W. Yang et al. / Physica B 403 (2008) 3009–30123010

on a second-rank tensor [1,16]. Moreover, for the MDsimulation, the time step is 1 fs, and the residual force peratom is 0.01 eV/A in the structural relaxations.

The calculated phonon dispersion relations (PDR) andvibrational density of states (VDOS) that yield the force-constant model are drawn in Figs. 1(a) and (b) for (10,10)armchair and (10,0) zigzag SWNTs, respectively. For 40carbon atoms in each unit cell, we have 120 vibrationaldegrees of freedom, but because of mode degeneracies thereare only 66 distinct phonon branches, of which 12 modesare non-degenerate and 54 are doubly degenerate. Thisresult is well described by non-symmorphic rod groups ofwhich the point group is D20h [17]. On the other hand, sincemany physical properties of solids, especially the first-orderRaman scattering and infrared absorption, depend on thePDR near q ¼ 0, we only consider the symmetry of thetube zone-center vibrations at the G-point (q ¼ 0). Conse-quently, the vibrational modes of the (10,10) and (10,0)SWNTs, respectively, are decomposed into the followingirreducible representations of D20h at the G-point [17]:

Gð10;10Þ ¼ 2A1g � 2A2g � 2B1g � 2B2g �A1u �A2u

� B1u � B2u � 2E1g � 4E2g � 2E3g

� 4E4g � . . .� ð3þ ð�1Þ9ÞE9g � 4E1u � 2E2u

� 4E3u � 2E4u � . . .� ð3� ð�1Þ9ÞE9u (1)

Gð10;0Þ ¼ 2A1g �A2g � 2B1g � B2g �A1u � 2A2u � B1u

� 2B2u �X9

j¼1

f3Ejg � 3Ejug (2)

Of these modes, the A1g, E1g, and E2g irreduciblerepresentations are Raman active, which is consistentwith the previous works [1,18–20] using the subgrouppoint group D10hDD20h. Due to the higher rod groupsymmetries, there are much fewer Raman-active modes(eight modes) than that of previous results (15–16 modes)[1,18–20]. It is important to note that the early worksperformed symmorphic space groups, i.e., omitted screwaxis operation, which results in the lower symmetry groupD10h.Even though group theory may indicate that a particular

mode is Raman active, this mode may nevertheless have asmall Raman cross-section [1]. In fact, only four Raman-active modes are strongly resonance enhanced experimen-tally. One of them, known as RBM, is dominated by thelower frequency A1g mode at around 200 cm�1, which isunique to CNTs without any counterpart in graphenesheets. The others, generally called TMs, are locatedaround 1600 cm�1 and correspond to the characteristicE2g modes of the graphene sheet. Furthermore, in themotions of RBM, not only for armchair tube but zigzag, allatoms move in the same way (in phase), perpendicular tothe tube axis, changing the radius of the tube, while in axialTMs [e.g. E1g mode in (10,10) tube; A1g and E2g modes in(10,0) tube] and circumferential TMs [e.g. A1g and E2g

modes in (10,10) tube; E1g mode in (10, 0) tube], two of thethree nearest-neighbor atoms move in opposite directions(out of phase) along and perpendicular to the tube axis.In fact, the CQC bond-stretching motions can be seenas horizontally and vertically vibrating CQC bond forarmchair and zigzag tubes, respectively.Altogether, these Raman-active modes depend strongly

on the tube symmetries. Once the tube structure istransformed, such as under hydrostatic pressure, the modeswill be certainly affected, and even the selection rulesmay be changed. If the modes are located at Ramanforbidden band, they will be no longer Raman active, andthus experimentally undetected by Raman spectra. Aredeformed modes still Raman active at much higherpressure? The analysis of the deformed structural symme-tries would be of particular importance for understandingthe essence of the RBM change under hydrostatic pressure.In order to explore a dependence of the RBM on the

symmetry alteration, on one hand, optimized structures ofthe (10,10) and (10,0) SWNTs under hydrostatic pressuresare simulated by the constant-pressure MD method. Thecross-section evolution of (10,10) and (10,0) SWNTs underselected pressure is shown in Figs. 2(a) and (b), respec-tively. Obviously, the pressure induces mechanical cross-section shape transition from a circle to a convex oval andthen to a non-convex oval shape. According to expecta-tions, the RBM frequency and its atomic motions vary withcircular-to-oval shape transition as shown in Fig. 3. Mostinterestingly, above a critical pressure PC [1.06GPa for the(10,10) tube and 5.50GPa for the (10,0) tube], the RBMstill exists and the atomic motions still exhibit the insetpattern of Fig. 3.

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On the other hand, group theory calculations arepresented for the RBM. From a careful analysis of thestructural data of deformations obtained by the above MDsimulation, the corresponding symmetry groups are at-tached in Fig. 2. Below the PC, a circular tube only shrinksby reducing its radius, so D20h symmetry still holds like inthe original tube without any pressure. Thereafter, it iseasier to bend than to compress a tube. At the PC,accordingly, the tube transforms from an analogous circleto an anisotropic oval shape, and then the symmetrylowers, with only three twofold rotational axes, to D2h

Fig. 2. Evolution of cross-sections of SWNTs under hydrostatic pressure,

obtained from MD simulations, for (a) (10,10) and (b) (10,0) SWNTs. The

corresponding pressure and symmetry group are attached.

0245

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ω (c

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ω (c

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(10, 0)

5.50 GPa

(10, 10)1.20GPa

1.10GPa

1.06GPa

0.90GPa

1 2 3 4 5 6 7Pressure (GPa)

Fig. 3. Calculated RBM frequency vs. pressure for (10,10) and (10,0)

SWNTs. The inset pattern shows atomic displacement patterns of RBM

for (10,10) SWNT at some selected pressure.

Table 1

Correlation of symmetry for D20h, D2h, and C2h point groups

D20h A1g A2g B1g B2g E1g E2g E3g

D2h Ag B1g Ag B1g B2g+B3g Ag+B1g B2g+B3g

C2h Ag Ag Ag Ag 2Bg 2Ag 2Bg

D20h A1u A2u B1u B2u E1u E2u E3u

D2h Au B1u Au B1u B2u+B3u Au+B1u B2u+B3u

C2h Au Au Au Au 2Bu 2Au 2Bu

point group. As the tube continues to shrink, it must adopta shape to minimize bending energy. This eventually leadsto another shape transition from a convex oval to a non-convex oval shape at 1.10 and 6.1GPa for (10,10) and(10,0) SWNTs, respectively. Meanwhile, the tube symmetrycontinues to lower, loose inversion centers, and degradedown to the C2h group. It should be noted that as theapplied pressure increases up to 1.25 and 7.0GPa for(10,10) and (10,0) SWNTs, respectively, the spacingbetween the opposite side walls approaches nearly 3.35 A,which means that an additional van der Waals (vdW)interaction may lead to the collapse of the tube [21], andthen the RBM may not be exhibited. Here, therefore, weonly focus on the (10,10) and (10,0) SWNTs subject tocompression up to 1.25 and 7.0GPa, respectively, withoutregarding the vdW interaction.Furthermore, to investigate the characteristic of new

symmetry modes in D2h and C2h groups, it is necessary toperform the correlation of irreducible representationsbetween D20h, D2h, and C2h point groups. The so-calledcorrelation is the relationship between a group and itssubgroups, which shows how the representations of agroup are changed or decomposed into those of itssubgroups when the symmetry is altered or lowered [22].In Table 1, the correlation of symmetry for D20h, D2h, andC2h point groups is charted, and the corresponding figure issketched in Fig. 4 (leaving out the associated ungerademode). With regard to the D20h point group, one of itsrepresentations, A1g, is changed to Ag mode for D2h andC2h point groups as shown in Fig. 4, while E1g and E2g aresplit: E1g-B2g+B3g-2Bg and E2g-Ag+B1g-2Ag.Moreover, corresponding to the RBM, the deformed Ag

modes of D2h and C2h point groups correlate to the modesof D20h, such as Ag-A1g+B1g+E2g+E4g+E6g+E8g.Since the tube symmetry is lowered, B1g, B2g, E4g, E6g,and E8g become Raman active, and then the deformedmodes are also Raman active. Consequently, above the PC,the deformed modes are still Raman active as supported bythe present theoretical calculations, though the RBM signalis experimentally undetectable [5–9] due to its Ramanintensity becoming too weak.In summary, using group theory calculations combined

with the force-constant model and MD simulations, wediscuss whether the deformed modes are Raman active ornot for (10,10) and (10,0) SWNTs under hydrostaticpressure. With increasing pressure, the tube symmetrylowers from D20h to D2h and then to C2h point group. For

E4g E5g E6g E7g E8g E9g

Ag+B1g B2g+B3g Ag+B1g B2g+B3g Ag+B1g B2g+B3g

2Ag 2Bg 2Ag 2Bg 2Ag 2Bg

E4u E5u E6u E7u E8u E9u

Au+B1u B2u+B3u Au+B1u B2u+B3u Au+B1u B2u+B3u

2Au 2Bu 2Au 2Bu 2Au 2Bu

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Fig. 4. Correlation figure between D20h, D2h, and C2h point groups,

leaving out ungerade mode associated.

W. Yang et al. / Physica B 403 (2008) 3009–30123012

D20h, A1g is changed to Ag, while E1g and E2g are split:E1g-B2g+B3g-2Bg and E2g-Ag+B1g-2Ag. On thebasis of the correlation between D20h, D2h, and C2h pointgroups, the deformed modes with D2h and C2h groups arestill Raman active. The intrinsic characteristics of thedeformed modes for an isolated SWNT under hydrostaticpressure have been shown by the present theoretical results,which may help us understand the essence of experimentalobservations.

This work is supported by PHR (IHLB), the NationalNatural Science Foundation of China (nos. 10604001 and60576012), and the Natural Science Foundation of Beijing(no. 4073029).

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